Maxwell's equations from divergence of stress-energy tensor?

[bcrowell]

If I start with the stress-energy tensor $T^{\mu\nu}$ of the electromagnetic field and then apply energy-momentum conservation $\partial_\mu T^{\mu\nu}=0$, I get a whole bunch of messy stuff, but, e.g., with $\nu=x$ part of it looks like $-E_x \nabla\cdot E$, which would vanish according to Maxwell's equations in a vacuum.

Is it true that you recover the complete vacuum version of Maxwell's equations by doing this? If so, is there any way to extend this to include the source terms?

 

[WannabeNewton]

$\nabla_{a}T^{ab} = -\frac{3}{2}F_{ac}\nabla^{[a}F^{bc]} + F_{c}{}{}^{b}\nabla_{a}F^{ac} =0$. From here you would have to somehow show that $\nabla^{[a}F^{bc]} = 0$ and $\nabla_{a}F^{ac} =0$. I don't immediately see a way to do that; even if you can show that the two surviving terms in $\nabla_a T^{ab} = 0$ are independent of each other, you'd still be left with $F_{ac}\nabla^{[a}F^{bc]} = 0$ and $F_{c}{}{}^{b}\nabla_{a}F^{ac} =0$.

 

[atyy]

If I am understanding MTW section 20.6 correctly, they say that Maxwell's equations can be derived from the Einstein field equation (G=T), which should be covariant conservation of energy, and the form of the stress-energy tensor.

But Exercise 20.8 is "The Maxwell equations cannot be derived from conservation of stress energy when (E.B) = 0 over an extended region".

 

[WannabeNewton]

If I am understanding MTW section 20.6 correctly, they say that Maxwell's equations can be derived from the Einstein field equation (G=T), which should be covariant conservation of energy, and the form of the stress-energy tensor.

Awesome find bud! They proceed directly from what I wrote down above to first show that $\nabla^{[a}F^{bc]} = 0$, which leaves $F_{c}{}{}^{b}\nabla_a F^{ac} = 0$ and they then argue that this can only vanish if $\nabla_a F^{ac} = 0$ by using invariants of the electromagnetic field. Their calculation is quite elegant.

 

[bcrowell]

Excellent -- thanks, atyy and WannabeNewton!

 

[WannabeNewton]

As for the source terms, usually one uses Maxwell's equations and the Maxwell stress-energy tensor to show that $\nabla_a T^{ab} = j_{a}F^{ab}$ but if you take this relation for granted and work backwards then you'd get $F_{c}{}{}^{b}\nabla_{a}F^{ac} = j^{c}F_{c}{}{}^{b}$; since this must hold for arbitrary electromagnetic fields you can easily conclude that $\nabla_a F^{ac} = j^c$. The only thing is that you can't use Maxwell's equations to prove that $\nabla_a T^{ab} = j_{a}F^{ab}$ like one normally does so if you want to work backwards you'd have to argue that $\nabla_a T^{ab} = j_{a}F^{ab}$ is true.

EDIT: and it's easy to argue this so as long as you assume that the total energy-momentum of the combined electromagnetic field + interacting charged fluid system is still conserved. Because if we have a charged fluid with some stress-energy tensor $T_{\text{mat}}^{ab}$ then we have $\nabla_a T_{\text{mat}}^{ab} = \mathcal{F}^b$, where $\mathcal{F}^b$ is the 4-force density on the charged fluid. Since the charged fluid is interacting with the electromagnetic field, the 4-force density comes directly from the Lorentz 4-force, whose density is simply $\mathcal{F}^b = -j_{a}F^{ab}$. So if the total energy-momentum is conserved then we will have $\nabla_a T^{ab}_{\text{em}} = j_a F^{ab}$.